1. The velocity components of a two-dimensional incompressible fluid flow are prescribed as u =-1 v=-3x²

i) Determine the stream function.

Step By Step Answers with Explanation

To do this, we can start by moving the e−2x−y term to the right side of the equation:

dy/dx = e^{2y} + e^{-2x-y}

dy/dx = e^{2y} / (e^{-2x-y} + 1)

Step 2: Integrate both sides

du / (u + 1) = dx

We can integrate both sides of this equation using the following formula:

Step 3: Solve for y

Now that we have integrated both sides of the equation, we can solve for y. To do this, we can start by exponentiating both sides of the equation:

Subtracting 1 from both sides and dividing both sides by ex, we get the following:

e^y = e^x e^C - 1

y = \ln (e^x e^C - 1)

where C is an arbitrary constant of integration.

C = \ln 2

Substituting this value of C into the general solution, we get the following particular solution: